We present some recent results on smooth vectors for unitary irreducible
representations of nilpotent Lie groups. Applications to the Weyl-Pedersen
calculus of pseudo-differential operators with symbols on the coadjoint orbits
are also discussed.

We survey some aspects of the pseudo-differential Weyl calculus for
irreducible unitary representations of nilpotent Lie groups, ranging from the
classical ideas to recently obtained results. The classical Weyl-H\"ormander
calculus is recovered for the Schr\"odinger representation of the Heisenberg
group. Our discussion concerns various extensions of this classical situation
to arbitrary nilpotent Lie groups and to some infinite-dimensional Lie groups
that allow us to handle the magnetic pseudo-differential calculus.